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-   -   Monday to Tuesday (https://www.mersenneforum.org/showthread.php?t=24147)

 wblipp 2019-03-05 23:54

Monday to Tuesday

This is really too simple for the puzzles section, but I think others might enjoy it.

My local gym has a monthly offering of "Pizza on the first Monday" and "Coffee and Bagels on the second Tuesday," These were consecutive days in January.

So how long is it, on average, from the first Monday to second Tuesday? How about the second Wednesday, Thursday, etc.?

 ATH 2019-03-06 03:34

[SPOILER]1st monday to 2nd tuedays is 8 days except when the month starts with tuesday, then it is 1 day.
In the 28 years cycle when weekdays match the same dates compared to leap years (not counting the year%100 and year%400 issues) there are actually exactly 1/7th of the months that starts with tuesdays. So on average (6/7)*8 + (1/7)*1 = 7 days.

Why is it in the 28 years (336 months) cycle the 1st of the months falls on a monday exactly 48 times, tuesday 48 times, ..., and sunday 48 times, even though the length of the months switches in the weird pattern between 28,29,30 and 31 days long ?[/SPOILER]

 Dr Sardonicus 2019-03-06 13:17

This reminds me of the cumbersome phrasing specifying Election Day here in the good ol' USA, which <google google> dates back to 1845, when a law was enacted mandating (my emphasis)[quote]That the electors of President and Vice President shall be appointed in each State [b]on the Tuesday next after the first Monday in the month of November[/b] of the year in which they are appointed[/quote]The first time I saw "first Tuesday after the first Monday" the effect on my mind was analogous to someone scraping their fingernails across a blackboard. "The Tuesday (next) after November 1" is much more shipshape.

 Uncwilly 2019-03-06 14:42

[QUOTE=Dr Sardonicus;510230]"The Tuesday (next) after November 1" is much more shipshape.[/QUOTE]The "next" is needed to be absolutely specific that it is not any Tuesday, but a specific one. Lawyers be like that you know.

 Dr Sardonicus 2019-03-06 15:02

[QUOTE=Uncwilly;510234]The "next" is needed to be absolutely specific that it is not any Tuesday, but a specific one. Lawyers be like that you know.[/QUOTE]Yes, they be. The fact that "The" is a definite (rather than an indefinite) article doesn't mean -- at least, not in a legally binding sense -- that it specifies the [i]first[/i] Tuesday after" or the Tuesday [i]next[/i] after November 1.

In fairness to lawyers, though, the insistence on what would seem to most people to be an unnecessary degree of specificity, almost certainly means that it actually [i]is[/i] necessary.

 wblipp 2019-03-07 22:12

I agree with ATH's calculation of average time from First Monday to Second Tuesday. How much longer is it to Second Wednesday?

 JeppeSN 2019-03-30 06:24

[QUOTE=ATH;510218]Why is it in the 28 years (336 months) cycle the 1st of the months falls on a monday exactly 48 times, tuesday 48 times, ..., and sunday 48 times, even though the length of the months switches in the weird pattern between 28,29,30 and 31 days long ?[/QUOTE]

In the Julian calendar, the leap year pattern repeats after 4 years which is 1,461 days. Now, when you consider weekdays, with 7 days in each week, you have to consider whether or not 7 divides 1,461. It does not. Therefore, the "least common multiple" where both leap year pattern and week day pattern repeats, will be 7 times 4 years, or in other words 7 times 1,461 days (that is 10,227 days). 7 is a prime.

So you do each of the 7 possible starts of the 1,461-day cycle. So any particular date (like the 1st of January, the 13rd of February, or anything) falls equally frequently on any day of week.

Let us see what happens in the Gregorian calendar. If we ignore the weekdays at first, the Gregorian calendar repeats after 400 years. 400 years is 146,097 (the 97-part is the 97 leap days in 400 years) in that system. Now, you would presuppose that you needed 7 times that period, i.e. 2,800 years, before both leap year and day-of-week pattern repeated, but that is wrong. Because, [B]by accident, 7 divides 146,097 (= 400*365 + 97)[/B]. In the Gregorian system, 400 years is exactly 20,871 weeks. Because of this coincidence, we do not get every possible date/day-of-week combination in the "expected" frequency.

2001-Jan-01 was a Monday. 2401-Jan-01 will be a Monday. 2801-Jan-01, 3201-Jan-01, etc. are all Mondays.

/JeppeSN

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